Does the integer k have a factor p such that 1 < p < k ?

This is not a value question. The question is NOT what is the value of p.

It's an YES/NO question: Does the integer k have a factor p such that 1<p<k? And from (2) we get a definite YES answer to that question.

Hope it's clear.

the question is asking whether k has a factor that is greater than 1, but less than itself.
if you're good at these number property rephrasings, then you can realize that this question is equivalent to "is k non-prime?", which, in turn, because it's a data sufficiency problem (and therefore we don't care whether the answer is "yes" or "no", as long as there's an answer), is equivalent to "is k prime?".
but let's stick to the first question - "does k have a factor that's between 1 and k itself?" - because that's easier to interpret, and, ironically, is easier to think about (on this particular problem) than the prime issue.

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key realization:
every one of the numbers 2, 3, 4, 5, ..., 12, 13 is a factor of 13!.

this should be clear when you think about the definition of a factorial: it's just the product of all the integers from 1 through 13. because all of those numbers are in the product, they're all factors (some of them several times over).

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consider the lowest number allowed by statement 2: 13! + 2.
note that 2 goes into 13! (as shown above), and 2 also goes into 2. therefore, 2 is a factor of this sum (answer to question prompt = "yes").

consider the next number allowed by statement 2: 13! + 3.
note that 3 goes into 13! (as shown above), and 3 also goes into 3. therefore, 3 is a factor of this sum (answer to question prompt = "yes").

etc.
all the way to 13! + 13.
works the same way each time.
so the answer is "yes" every time --> sufficient.

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Bunuel - would it be correct to say that all factorials are composite numbers?

Yes, but except 0! = 1, 1! = 1, and 2! = 2.

If the answer is NO -- if k does NOT have a factor p between 1 and k -- the implication is that k is PRIME.
Question stem, rephrased:
Is k prime?

Statement 1:
Clearly, many values greater than 4! will be prime, while most will not be prime.
Thus, the answer to the question stem can be YES or NO.
INSUFFICIENT.

Statement 2:
Since the GMAT cannot expect us to prove that any of the integers within the given range ARE prime, all of the integers within the given range must NOT be prime.
Thus, the answer to the question stem is NO.
SUFFICIENT.

Bunuel could you please confirm can i use the below mentioned approach for the option 2?

(2) 13!+2≤k≤13!+13

Subtract 13! from each sides

2≤k-13!≤ 13
Now k must be 16!
so now there should be a 5*2 pair in k thus it is not a prime number .

No this is not correct. If k = 16!, then 16! - 13! is MUCH larger than 13.

thank you Bunuel and sorry for asking a dumb question just realised how silly my question was .

If integer k has a factor p such that 1<p<k, then the integer is composite. If it doesn’t, then k is prime.
We can therefore rephrase the question as “Is k prime?”

From statement I alone, k > 4! i.e. k > 24.

This is not sufficient to find out if k is prime, because k could be any integer value greater than 24.
Statement I alone is insufficient. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, 13! + 2 ≤ k ≤ 13! + 13.

It’s important to note that 13! will be divisible by all integers from 1 to 13. Therefore, regardless of the value of k, k will be divisible by at least one more integer other than 1 and itself.

k is NOT prime.
Statement II alone is sufficient to answer the question with a YES. Answer options C and E can be eliminated,

The correct answer option is B.

Hope that helps!
Aravind B T

Help would be appreciated! (+Kudos if you had the same issue)
1) the question asked whether p is a factor of k, well with statement (B) if we assume p=17 (17 is less than 13!+2 so p=17 could be assumed) then p wont be a factor of k (because the largest prime k has is 13) , and if p=18 it would be a factor so we were not able to determine whether or not k has a factor of p (as it depends on the value of p),
2) the question hadnt mentioned whether p is an integer either (if p=2 then yes its a factor of k, if p = 1.5 then its not)
Many thanks!!

Hey Bunuel, why did you multiple #2 by 8 if it was just adding 8?

For example if it said 5! + 11 it would be 120 +11 which would become a prime number but if it was 5! +10, it would not be a prime number

We did not multiply by 8, we factored out 8 from 13! + 8 = 2*3*4*5*6*7*8*9*10*11*12*13 + 8 to get 8*(2*3*4*5*6*7*9*10*11*12*13+1).

Hope it helps.

Thanks very much for the clarification Bunuel!